Classification of Links Up to 0-Solvability

TL;DR

This paper characterizes 0-solvable links within the $n$-solvable filtration, introduces a new equivalence relation, and classifies the set of such links using algebraic and geometric methods, connecting to low-dimensional topology.

Contribution

It introduces 0-solve equivalence, provides an algebraic and geometric classification of 0-solvable links, and establishes their structure as a specific quotient group.

Findings

The set of 0-solvable links forms a group isomorphic to a specific quotient of concordance classes.

0-solvable links are exactly those bounding class 2 gropes and supporting order 2 Whitney towers.

The group of 0-solvable links is isomorphic to ^m  ^{m }.

Abstract

The $n$-solvable filtration of the $m$-component smooth (string) link concordance group, $$\dots \subset \mathcal{F}^m_{n+1} \subset \mathcal{F}^m_{n.5} \subset \mathcal{F}^m_n \dots \subset \mathcal{F}^m_1 \subset \mathcal{F}^m_{0.5} \subset \mathcal{F}^m_0 \subset \mathcal{F}^m_{-0.5} \subset \mathcal{C}^m,$$ as defined by Cochran, Orr, and Teichner, is a tool for studying smooth knot and link concordance that yields important results in low-dimensional topology. The focus of this paper is to give a characterization of the set of 0-solvable links. We introduce a new equivalence relation on links called 0-solve equivalence and establish both an algebraic and a geometric classification of $\mathbb{L}_0^m$, the set of links up to 0-solve equivalence. We show that $\mathbb{L}_0^m$ has a group structure isomorphic to the quotient $\mathcal{F}_{-0.5}/\mathcal{F}_0$ of concordance classes of string links and classify this group, showing that $$\mathbb{L}_0^m \cong \mathcal{F}_{-0.5}^m/\mathcal{F}_0^m \cong \mathbb{Z}_2^m \oplus \mathbb{Z}^{m \choose 3} \oplus \mathbb{Z}_2^{m \choose 2}.$$ Finally, using results of Conant, Schneiderman, and Teichner, we show that 0-solvable links are precisely the links that bound class 2 gropes and support order 2 Whitney towers in the 4-ball.